SSC CGL Maths Arithmetic Ratio & Proportion

Ratio and Proportion form the foundation of many arithmetic questions asked in SSC CGL, SSC CHSL, CPO, and other government exams. A ratio compares two quantities, while a proportion shows the equality of two ratios. Although the basic concepts are simple, competitive exams often test candidates through advanced applications and shortcut techniques.

To score well, candidates should understand how to derive ratios from equations, simplify fractional ratios, solve chain ratio problems, and apply special methods for finding unknown values. With regular practice and a clear understanding of these concepts, Ratio and Proportion can become one of the easiest and highest-scoring topics in the quantitative aptitude section.

Ratio and Proportion

Ratio and Proportion is one of the most important topics in arithmetic and is frequently asked in competitive exams such as SSC CGL and SSC CHSL. While the basic concepts are simple, exam questions often require smart problem-solving techniques and a strong understanding of ratio relationships. 

 Learning methods such as direct substitution, chain ratios, compound ratios, and shortcut approaches can help candidates solve questions more quickly and accurately. A clear grasp of these concepts, combined with regular practice, can significantly improve performance in the quantitative aptitude section.

Direct Substitution in Ratios

When given a direct ratio between two variables, such as x:y = 4:5, you can directly substitute these ratio values into an expression to find the resulting ratio. This method is applicable when the expression is homogeneous, meaning all terms or numerator/denominator parts have the same degree for the variables.

 Problem Example: If x:y is 4:5, find the value of (8x - 6y) : (9x - 7y).

 Solution Strategy:

1. Substitute x = 4 and y = 5.

2. Numerator: (8 * 4) - (6 * 5) = 32 - 30 = 2.

3. Denominator: (9 * 4) - (7 * 5) = 36 - 35 = 1.

4. Resulting Ratio: 2:1.

Ratio from Algebraic Equation

Often, ratios need to be derived from a given algebraic equation before solving for another expression.

 Problem Example: Given (10a + 4b) / (11a - 15b) = 7/5, find the ratio of (5a + 3b) / (9a - 2b).

 Solution Strategy:

1. Derive a:b Ratio:

• Cross-multiply: 5(10a + 4b) = 7(11a - 15b)

• 50a + 20b = 77a - 105b

• 125b = 27a

• This gives a/b = 125/27. Recognizing 125 as 5³ and 27 as 3³, this simplifies to a:b = 5:3.

2. Substitute a = 5 and b = 3 into the target expression:

• Numerator: (5 * 5) + (3 * 3) = 25 + 9 = 34.

• Denominator: (9 * 5) - (2 * 3) = 45 - 6 = 39.

3. Resulting Ratio: 34/39.

Fractional Ratios (A:B:C)

When ratios are given in fractional form, simplify them to whole numbers first, then combine. Using options is a highly recommended strategy.

 Problem Example: Given A:B = 1/3 : 1/5 and B:C = 1/7 : 1/8, find A:B:C.

 Solution Strategy:

1. Simplify A:B: Cross-multiply (1 * 5) : (1 * 3) = 5:3.

2. Simplify B:C: Cross-multiply (1 * 8) : (1 * 7) = 8:7.

3. Use Options: Check the given options. The correct option will have A:B as 5:3 and B:C as 8:7. When looking for A:B, ensure A's component is a multiple of 5 and B's is a multiple of 3. Often, one option satisfies this.

• For example, an option 40:24:21:

• A:B = 40:24 = 5:3 (dividing by 8)

• B:C = 24:21 = 8:7 (dividing by 3)

Multi-Step Chain Ratio (A:B:C:D:E)

For complex chain ratios, the options method is the most efficient.

 Solution Strategy:

1. Identify the first given ratio (e.g., A:B = 4:7).

2. Filter options: Check which overall ratio option (A:B:C:D:E) has its A and B components simplifying to the required 4:7. This often eliminates most incorrect options quickly.

Compound Ratios using "Hide-and-Seek" Method

This method quickly finds combined ratios when terms are equated (e.g., 3A = 4B = 5C).

 Solution Strategy:

1. To find A's ratio: Hide the 'A' term (3A) and multiply the coefficients of the other terms (4 * 5 = 20).

2. To find B's ratio: Hide the 'B' term (4B) and multiply the coefficients of the other terms (3 * 5 = 15).

3. To find C's ratio: Hide the 'C' term (5C) and multiply the coefficients of the other terms (3 * 4 = 12).

4. Resulting Ratio: A:B:C = 20:15:12.

• (Memory Tip: The "Hide-and-Seek" method is like "I Spy" – what you're looking for, you hide, and combine (multiply) the remaining coefficients.)

Direct Calculation of A:D in Chain Ratios

When given a chain of ratios like A:B, B:C, C:D, you can directly find the A:D ratio without calculating the full A:B:C:D.

 Method:

• To find the 'A' component: Multiply the first parts of all individual ratios.

• To find the 'D' component: Multiply the second parts of all individual ratios.

Example Problem: Given A:B = 3:4, B:C = 5:7, C:D = 8:3. Find A:D.

 Solution:

1. A component: 3 * 5 * 8 = 120

2. D component: 4 * 7 * 3 = 84

3. A:D ratio: 120:84

4. Simplify (divide by 12): A:D = 10:7.

Conceptual Understanding: This shortcut works because (A/B) * (B/C) * (C/D) = A/D. Multiplying the numerators gives A's part, and multiplying the denominators gives D's part.

Chain Rule for Ratios with Multipliers (2A:3D)

This involves finding a specific ratio like xA:yD from a chain.

Problem Example: Given A:B = 3:5, B:C = 7:8, C:D = 2:3. Find the ratio of 2A:3D.

Solution Strategy:

1. Find A:D using direct multiplication:

• A component: 3 * 7 * 2

• D component: 5 * 8 * 3

• So, A:D = (3 * 7 * 2) : (5 * 8 * 3)

2. Apply multipliers and simplify:

• Target: 2A : 3D

• (2 * 3 * 7 * 2) : (3 * 5 * 8 * 3)

• Cancel common terms (e.g., '3' from both sides): (2 * 7 * 2) : (5 * 8 * 3)

• Further simplify: (4 * 7) : (5 * 24) = 28 : 120

• Divide by 4: 7:30.

Sum Equal Method for Ratios: (a+b)/c Form

When ratios are given in forms like (a+b)/c, and they relate to the same total sum (a+b+c), it's crucial to equalize the total sum of the ratio parts across all given ratios.

 Problem Example: Given (a+b)/c = 23/12 and (b+c)/a = 17/8. Find (a+c)/b.

 Solution Strategy:

1. Calculate total sum for each ratio:

• From (a+b)/c = 23/12, total (a+b+c) = 23 + 12 = 35.

• From (b+c)/a = 17/8, total (a+b+c) = 17 + 8 = 25.

2. Equalize these sums: Find the LCM of 35 and 25, which is 175.

• Multiply first ratio by (175/35) = 5.

• Multiply second ratio by (175/25) = 7.

3. Adjust ratios:

• (a+b)/c = (23*5)/(12*5) = 115/60. So, (a+b) = 115, c = 60.

• (b+c)/a = (17*7)/(8*7) = 119/56. So, (b+c) = 119, a = 56.

4. Extract individual values (a, b, c):

• We have a+b+c = 175.

• a = 56, c = 60.

• b = (a+b+c) - a - c = 175 - 56 - 60 = 175 - 116 = 59.

• So, a=56, b=59, c=60.

5. Calculate target expression (a+c)/b:

• (56 + 60) / 59 = 116/59.

Ratio of 1/A:1/B:1/C from (A+B), (B+C), (C+A)

This is a very important type involving sums of two variables.

 Problem Example: Given (a+b)=7, (b+c)=6, (c+a)=5 (in ratio terms). Find 1/a : 1/b : 1/c.

 Solution Strategy:

1. Find the total sum (a+b+c) in ratio terms:

• Add all given ratio parts: (a+b)+(b+c)+(c+a) = 7+6+5 = 18.

• This equals 2a+2b+2c = 2(a+b+c).

• So, 2(a+b+c) = 18, meaning (a+b+c) = 9.

2. Extract individual ratio values (a, b, c):

• a = (a+b+c) - (b+c) = 9 - 6 = 3.

• b = (a+b+c) - (c+a) = 9 - 5 = 4.

• c = (a+b+c) - (a+b) = 9 - 7 = 2.

• Thus, a:b:c = 3:4:2.

3. Calculate reciprocal ratio 1/a : 1/b : 1/c:

• 1/3 : 1/4 : 1/2.

4. Convert to whole numbers: Find LCM of denominators (3, 4, 2) which is 12.

• Multiply each fraction by 12: (1/3)*12 : (1/4)*12 : (1/2)*12 = 4:3:6.

Finding 3A + B - 4C from Sums and a Total Value

This problem integrates ratio calculations with actual values.

 Problem Example: Given (a+b):(b+c):(c+a) = 15:14:11 (in ratio terms). Also, (a+b+c) = 40 (actual value). Find the value of (3a + b - 4c).

 Solution Strategy:

1. Find (a+b+c) in ratio terms:

• Sum of parts: 15+14+11 = 40.

• This is 2(a+b+c) = 40. So, (a+b+c) = 20 (ratio units).

2. Extract individual ratio values (a, b, c):

• c = (a+b+c) - (a+b) = 20 - 15 = 5.

• a = (a+b+c) - (b+c) = 20 - 14 = 6.

• b = (a+b+c) - (c+a) = 20 - 11 = 9.

• Thus, a:b:c = 6:9:5 (ratio units).

3. Relate ratio units to actual value:

• Total ratio units (a+b+c) = 6+9+5 = 20 units.

• Given actual sum = 40.

• So, 20 ratio units = 40.

• Therefore, 1 ratio unit = 2.

4. Calculate actual values:

• a = 6 * 2 = 12.

• b = 9 * 2 = 18.

• c = 5 * 2 = 10.

5. Calculate target expression (3a + b - 4c):

• (3 * 12) + (18) - (4 * 10) = 36 + 18 - 40 = 54 - 40 = 14.

Complex Ratio Multiplication (ABC: DEF)

This problem type requires understanding how to derive a product ratio from a series of individual ratios. This pattern is very important.

 Concept: If you need to find a ratio like (A*B*C)/(D*E*F), you can achieve this by multiplying a series of appropriate individual ratios, allowing for cancellations.

 Solution Strategy (General):

1. Identify the target product ratio.

2. Multiply the relevant individual ratios. For example, if you have A/D, B/E, C/F, then (A/D) * (B/E) * (C/F) will give (A*B*C)/(D*E*F).

3. Perform cancellations to simplify the product to the final ratio.